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help-Continuity

0 votes

asked Apr 27, 2015 in CALCULUS by anonymous

8 Answers

0 votes

(2_a)

Step 1:

The function is and .

Definition of continuity :

A function is continuous at , if then it should satisfy three conditions :

(1) is defined.

(2) exists.

(3) .

Condition (1): is defined.

The function is .

Substitute in the function.

.

is defined.

Step 2:

Condition (2): exists.

.

exists.

answered Apr 27, 2015 by joseph Apprentice

Contd....

Step 3:

Condition (3): .

and .

.

The three conditions of continuity are satisfied, hence the function is continuous.

The function is continuous at .

Solution :

The function is continuous at .

0 votes

(2_b)

Step 1:

The function is and .

Definition of continuity :

A function is continuous at , if then it should satisfy three conditions :

(1) is defined.

(2) exists.

(3) .

Condition (1): is defined.

The function is .

Substitute in the function.

.

is defined.

Step 2:

Condition (2): exists.

image.

exists.

answered Apr 27, 2015 by joseph Apprentice
edited Apr 27, 2015 by joseph

Contd....

Step 3:

Condition (3): .

image and .

.

The three conditions of continuity are satisfied, hence the function is continuous.

The function is continuous at .

Solution :

The function is continuous at .

0 votes

(2_c)

Step 1:

The function is and .

Definition of continuity :

A function is continuous at , if then it should satisfy three conditions :

(1) is defined.

(2) exists.

(3) .

Condition (1): is defined.

The function is .

Substitute in the function.

is undefined.

The condition (1) of continuity is not satisfied, hence the function is discontinuous.

The function is discontinuous at .

Solution :

The function is discontinuous at .

answered Apr 27, 2015 by joseph Apprentice
0 votes

(3_a)

Step 1:

The function is and .

Definition of continuity :

A function is continuous at , if then it should satisfy three conditions :

(1) is defined.

(2) exists.

(3) .

Condition (1): is defined.

If then .

Substitute in above function.

.

is defined.

Step 2:

Condition (2): exists.

.

exists.

answered Apr 27, 2015 by joseph Apprentice

Contd....

Step 3:

Condition (3): .

and .

.

The three conditions of continuity are satisfied, hence the function is continuous.

The function is continuous at .

Solution :

The function is continuous at .

0 votes

(3_b)

Step 1:

The function is and .

Definition of continuity :

A function is continuous at , if then it should satisfy three conditions :

(1) is defined.

(2) exists.

(3) .

Condition (1): is defined.

If then .

Substitute in above function.

.

is defined.

Step 2:

Condition (2): exists.

Left hand limit :

If then .

.

Right hand limit :

If then .

.

answered Apr 27, 2015 by joseph Apprentice

Contd....

Left hand limit and right hand limit are not equal, limit does not exist.

does not exists.

Here limit does not exist at .

The condition (2) of continuity is not satisfied, hence the function is discontinuous.

The function is discontinuous at .

Solution :

The function is discontinuous at .

0 votes

(4_a)

Step 1:

The function is and .

Simplify the function.

Definition of continuity :

A function is continuous at , if then it should satisfy three conditions :

(1) is defined.

(2) exists.

(3) .

Condition (1): is defined.

The function is .

Substitute in the function.

is undefined.

The condition (1) of continuity is not satisfied, hence the function is discontinuous.

The function is discontinuous at .

Solution :

The function is discontinuous at .

answered Apr 27, 2015 by joseph Apprentice
0 votes

(4_b)

Step 1:

The function is and .

Simplify the function.

Definition of continuity :

A function is continuous at , if then it should satisfy three conditions :

(1) is defined.

(2) exists.

(3) .

Condition (1): is defined.

The function is .

Substitute in the function.

.

 is defined.

Step 2:

Condition (2): exists.

.

exists.

answered Apr 27, 2015 by joseph Apprentice

Contd....

Step 3:

Condition (3): .

and .

.

The three conditions of continuity are satisfied, hence the function is continuous.

The function is continuous at .

Solution :

The function is continuous at .

0 votes

(4_c)

Step 1:

The function is and .

Simplify the function.

Definition of continuity :

A function is continuous at , if then it should satisfy three conditions :

(1) is defined.

(2) exists.

(3) .

Step 2:

Condition (1): is defined.

The function is .

Substitute in the function.

is undefined.

The condition (1) of continuity is not satisfied, hence the function is discontinuous.

The function is discontinuous at .

Solution :

The function is discontinuous at .

answered Apr 27, 2015 by joseph Apprentice

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